\(q\)-Taylor theorems, polynomial expansions, and interpolation of entire functions. (Q1399540)

The authors establish \(q\)-analogues of Taylor series expansions in special polynomial bases for functions analytic in bounded domains and for entire functions whose maximum modulus \(M(r;f)\) satisfies \(| \ln M(r;f)| \leq A \ln^2 r\). This solves the problem of constructing such entire functions from their values at \([aq^n +q^{-n}/a]/2\), for \(0<q<1\). Their technique is constructive and gives an explicit representation of the sought entire function. Applications to \(q\)-series identities are given.